A248815 Numbers equal to the arithmetic derivative of their Euler totient function.

31, 32, 80, 81, 112, 176, 192, 244, 368, 752, 859, 912, 944, 1296, 1328, 1712, 1723, 2672, 2864, 3024, 3632, 4208, 5552, 5744, 6128, 6156, 7472, 7664, 8048, 8748, 9008, 9392, 11504, 13424, 13808, 14192, 15728, 16304, 18992, 20412, 20528, 20912, 21104, 21872

Offset: 1

Author

Paolo P. Lava, Oct 15 2014

Keywords

• nonn

Comments

Solutions of the equation n = (phi(n))’.

Examples

			Euler totient function of 32 is 16 and the arithmetic derivative of 16 is 32.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Paolo P. Lava)

Crossrefs

Cf. A000005, A003415, A166374.

Programs

Maple

with(numtheory): P:= proc(q) local a,n,p; for n from 1 to q do
a:=phi(n)*add(op(2,p)/op(1,p),p=ifactors(phi(n))[2]);
if n=a then print(n); fi; od; end: P(10^9);

Mathematica

deriv[n_] := n*Total[#2/#1 & @@@ FactorInteger[n]]; aQ[1] = 1; aQ[n_] := deriv[EulerPhi[n]] == n; Select[Range[25000], aQ] (* Amiram Eldar, Jul 11 2019 *)